Affiliation:
1. Polytechnic Institute of NYU
2. TU Eindhoven, MB Eindhoven, The Netherlands
3. Utrecht University, TB Utrecht, The Netherlands
4. Utrecht University
Abstract
In the trajectory segmentation problem, we are given a polygonal trajectory with
n
vertices that we have to subdivide into a minimum number of disjoint segments (subtrajectories) that all satisfy a given criterion. The problem is known to be solvable efficiently for
monotone
criteria: criteria with the property that if they hold on a certain segment, they also hold on every subsegment of that segment. To the best of our knowledge, no theoretical results are known for nonmonotone criteria.
We present a broader study of the segmentation problem, and suggest a general framework for solving it, based on the
start-stop diagram
: a 2-dimensional diagram that represents all valid and invalid segments of a given trajectory. This yields two subproblems: (1) computing the start-stop diagram, and (2) finding the optimal segmentation for a given diagram. We show that (2) is NP-hard in general. However, we identify properties of the start-stop diagram that make the problem tractable and give a polynomial-time algorithm for this case.
We study two concrete nonmonotone criteria that arise in practical applications in more detail. Both are based on a given univariate attribute function
f
over the domain of the trajectory. We say a segment satisfies an
outlier-tolerant criterion
if the value of
f
lies within a certain range for at least a given percentage of the length of the segment. We say a segment satisfies a
standard deviation criterion
if the standard deviation of
f
over the length of the segment lies below a given threshold. We show that both criteria satisfy the properties that make the segmentation problem tractable. In particular, we compute an optimal segmentation of a trajectory based on the outlier-tolerant criterion in
O
(
n
2
log
n
+
kn
2
) time and on the standard deviation criterion in
O
(
kn
2
) time, where
n
is the number of vertices of the input trajectory and
k
is the number of segments in an optimal solution.
Funder
EU Cost Action IC0903
NSF
NSA MSP
Netherlands Organisation for Scientific Research
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
14 articles.
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